It forms the heart of the proof via Ricci flow of Thurston's Geometrization Conjecture. Thurston's Geometrization Conjecture, which classifies all compact 3-manifolds, will be the subject of a follow-up article. The organization of the material in this book differs from that given by Perelman. From the beginning the authors present all analytic and geometric arguments in the context of Ricci flow with surgery. In addition, the fourth part is a much-expanded version of Perelman's third preprint; it gives the first complete and detailed proof of the finite-time extinction theorem.
With the large amount of background material that is presented and the detailed versions of the central arguments, this book is suitable for all mathematicians from advanced graduate students to specialists in geometry and topology. The Clay Mathematics Institute Monograph Series publishes selected expositions of recent developments, both in emerging areas and in older subjects transformed by new insights or unifying ideas. The comprehensive and carefully detailed nature of the text makes this book an invaluable resource for any mathematician who wants to understand the technical nuts and bolts of the proof, while the introductory chapter provides an excellent conceptual overview of the entire argument.
Continue Shopping Checkout. AMS Homepage. Join our email list. Sign up. Advanced search. Volume: 3. Publication Month and Year: Copyright Year: Page Count: Cover Type: Hardcover. Print ISBN Print ISSN: Primary MSC: 53 ; Ultimately, Russian mathematician Grigory Perelman solved this conjecture in the affirmative in He was awarded Fields Medal in In this talk, I wish to give some background of geometry of Riemannian 3-manifolds, and then describe briefly some glimpses of this highly technical work.
Perelman used many highly innovative techniques in his proof which will surely pave new directions in the study of relationship between topology and geometry of three dimensional manifolds.
Poincare announced this conjecture in , and since then leading mathematicians put their efforts in proving or disproving it. One possible reason for this is the following : The fundamental group plays an important role in the analysis of these topologies and the relations between generators of the fundamental group correspond to two dimensional disks, mapped into the manifold.
In dimensions greater than or equal to five, such disks can be put into positions so that they are disjoint from each other, with no self- intersections, but in dimensions three or four, it may not be possible to avoid intersections. This leads to serious difficulties with respect to embeddings. He gave complete classification of all closed simply-connected topological 4-manifolds. Only three dimensional case remained to be solved.
However, there are technical issues associated with this program: In general Ricci flow will develop singularities in finite time and thus we must have a method for analyzing these singularities and continuing the flow past them. Hamilton made significant progress by establishing many important and useful analytic estimates to understand the evolving metric and its curvatures. Thus the manifold under consideration has a hyperbolic metric. The work is quite intricate and involves many original ideas.
The detailed analysis can be found in Cao and Zhu [ 6 ], and Morgan and Tian [ 7 ]. This lecture article is a small and modest effort to introduce the audience readers to the work of Hamilton and Perelman, and state important results proved by them.
The author does not claim any originality. We also refer the reader to other expository articles by Gadgil and Sheshadri [ 8 ], Milnor [ 9 ] and Kominers [ 10 ]; and to review articles by Morgan [ 11 ] and Milnor [ 12 ].
The recent book by Morgan and Tian [ 7 ] is a very useful reference to this subject. We assume that the reader has a basic knowledge of Riemannian Geometry, cf. Kobayashi and Nomizu — Vol. It is well-known that if M is simply-connected, then its fundamental group which is also called the first homotopy group is trivial.
For example, S2 unit sphere in R3 has this property, whereas T2 two dimensional torus does not possess this property. Thus, Poincare Conjecture in three dimensions states that a compact, connected, simply-connected 3-manifold is diffeomorphic to S3.
Let u and v be tangent vectors to M at p. The sectional curvature determines the Curvature Tensor completely. It is also called hyperbolic manifold. It can be thought of as the negative curvature analogue of the n-sphere Sn.
Next, consider a Lie group G. Hence any inner product on TeG induces a left-invariant Riemannian metric on G i. Also it is true that any Riemannian metric on G for which all left translations are Riemannian isometries is of this form i. Connected sum : Given two surfaces S1 and S2, the connected sum S1 S2 is constructed by removing a disk from each and then joining them along the boundaries of the holes. For example, the connected sum of two tori is a 2-holed torus. If M is a prime 3-manifold, then either it is S2 x S1 or the non-orientable S2- bundle over S1 or any embedded 2-sphere in M bounds a ball, i.
The prime decomposition theorem for 3-manifolds states that every compact, orientable 3-manifold is the connected sum of a unique collection of Prime 3- manifolds. An n-manifold is called irreducible if any n-1 - sphere bounds an embedded n-ball. Irreducible manifold is prime, but the converse is not true. The manifold is called sufficiently large if it contains an incompressible surface.
Homogeneous Geometries : A homogeneous Riemannian manifold M, g is one whose group of isometries acts transitively on the manifold.
The Riemannian metric on Hn is the one induced by restricting Q to the tangent planes to the hypersurface. The subgroup of index two in the orthogonal group of Q consisting of all transformations that leave invariant the upper sheet of hyperbola acts transitively as the full group of isometries of Hn. All sectional curvatures of Hn are equal to We say that a Riemannian manifold is modeled on a given homogeneous manifold M, g if every point of the manifold has a neighbourhood which is isometric to an open set in M, g.
If such manifolds are complete, they are called Locally Homogeneous Manifolds. The 3-sphere S3 ii. The Euclidean space R3 iii. The Hyperbolic space H3 iv. The direct product of a 2-sphere and a circle, S2 x S1 v.
The direct product of a hyperbolic plane and a circle, H2 x S1 vi. A left invariant Riemannian metric on the special linear group SL 2, R vii.
A left invariant Riemannian metric on the Poincare-Lorentz group viii. Any locally homogeneous manifold modeled on the first is a Riemannian manifold of constant positive sectional curvature. These manifolds are called spherical space-forms. A locally homogeneous manifold modeled on the second are flat 3-manifolds.
Compact flat 3-manifolds are flat tori. A locally homogeneous manifold modeled on H3 is a complete hyperbolic manifold with constant sectional curvature equal to Geometries in iv and v above are homogeneous 3-manifolds with product metrics S2 x R and H2 x R respectively. Last three manifolds are homogeneous manifolds M, g where M is a simply-connected Lie group and g is a left invariant metric. The group acts isometrically on itself by left multiplication and hence is embedded in the group of isometries of M.
Locally homogeneous manifolds modeled on Heisenberg group are compact if they are finite volume, and are called nil-manifolds. Those modeled on Poincare-Lorentz group are also compact and are called solv-manifolds.
Similarly, locally homogeneous manifolds modeled on SL 2, R are also compact. As two special cases, which the above conjecture would imply, we have : I. A closed 3-manifold has finite fundamental group if and only if it has a metric of constant positive curvature.
In particular, any 3-manifold M with trivial fundamental group must be homeomorphic to S3. In the special case of a manifold which is sufficiently large, Thurston himself proved this last statement and in fact proved the full Geometrization Conjecture. The remaining six geometries are well understood. See Thurston [ 3 ] and Scott [ 15 ] for detailed expositions. This argument also applies to prime 3-manifolds of finite fundamental group. If I has an initial point t0, then M, g t0 is called the initial condition of or the initial metric for the Ricci flow or of the solution.
In simple situations, the flow can be used to deform g into a metric distinguished by its curvature. For example, if M is 2-dimensional, the Ricci flow, once suitably renormalized, deforms a metric conformally to a metric of constant curvature, and thus gives a proof of the two dimensional Uniformisation Theorem. Two dimensional Uniformisation Theorem states that If X is a compact surface, then X admits a locally homogeneous metric locally isometric to one of the constant curvature models, spherically symmetric, flat or hyperbolic, depending on whether the curvature is positive, zero or negative.
More generally, the topology of M may avoid the existence of such distinguished metrics, and the Ricci flow can then be expected to develop a singularity in finite time. However, the behaviour of the flow may still give the information about the topology of the underlying manifold.. If we understand the structure of finite time singularities sufficiently well, then we may keep track of the change in topology of the manifold under surgery, and reconstruct the topology of the original manifold from a limiting flow, together with the singular regions removed.
In harmonic coordinates x1, x2, … , xn about p, i. This is a one-parameter family of metrics. To relate Einstein metric to the Poincare Conjecture, we note that an Einstein metric g on a 3-manifold necessarily has constant sectional curvature. In fact, in all dimensions, metrics of constant sectional curvature are Einstein metrics. Hence, we conclude that if M, g is closed, simply-connected and Einstein, then M, g is isometric to S3.
Euclidean and hyperbolic spaces are ruled out because they are not closed. Hence the Poincare Conjecture can be formulated as saying that any closed, simply-connected 3-manifold has an Einstein metric.
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